This paper deals with a Cahn-Hilliard equation with variable exponent sources. By using the potential well method, we give some threshold results on existence and nonexistence of global weak solutions when initial data with energy less than the potential well depth $d$. In the former case, we also show the exponential decay properties of energy functional. We finally obtain some sufficient conditions for the global existence and non-global existence results with high energy initial data. The results of this paper extend some recent results of Han (2018) \cite{Han18} and Zhou (2019) \cite{Zhou19} to the case of PDEs with variable exponent sources.

This paper deals with a pseudo-parabolic equation involving variable exponents under Dirichlet boundary value condition. The authors first develop the potential well method to prove a threshold result on the existence and nonexistence of global solutions to the equations when initial energy is less than the mountain pass level $d$. By borrowing the idea from \cite{GW05,XS13} we also show some existence and nonexistence results with high energy initial data. In this case a new characterization for nonexistence of solution is given. These results extend and improve a recent result obtained by Di et al. (2017) \cite{DSP17} and Liao (2019) \cite{Liao19,LGL19}.